• Bernoulli
  • Volume 23, Number 4A (2017), 2917-2950.

Guided proposals for simulating multi-dimensional diffusion bridges

Moritz Schauer, Frank van der Meulen, and Harry van Zanten

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A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form. We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.

Article information

Bernoulli Volume 23, Number 4A (2017), 2917-2950.

Received: September 2014
Revised: October 2015
First available in Project Euclid: 9 May 2017

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change of measure data augmentation linear processes multidimensional diffusion bridge


Schauer, Moritz; van der Meulen, Frank; van Zanten, Harry. Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli 23 (2017), no. 4A, 2917--2950. doi:10.3150/16-BEJ833.

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  • [1] Aronson, D.G. (1967). Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. (N.S.) 73 890–896.
  • [2] Bayer, C. and Schoenmakers, J. (2014). Simulation of forward-reverse stochastic representations for conditional diffusions. Ann. Appl. Probab. 24 1994–2032.
  • [3] Beskos, A., Papaspiliopoulos, O. and Roberts, G.O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077–1098.
  • [4] Beskos, A., Roberts, G., Stuart, A. and Voss, J. (2008). MCMC methods for diffusion bridges. Stoch. Dyn. 8 319–350.
  • [5] Beskos, A. and Roberts, G.O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422–2444.
  • [6] Bladt, M. and Sørensen, M. (2014). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Bernoulli 20 645–675.
  • [7] Chicone, C. (1999). Ordinary Differential Equations with Applications. Texts in Applied Mathematics 34. New York: Springer.
  • [8] Clark, J. (1990). The simulation of pinned diffusions. In Proceedings of the 29th IEEE Conference on Decision and Control, 1990 1418–1420. Piscataway, NJ: IEEE.
  • [9] Delyon, B. and Hu, Y. (2006). Simulation of conditioned diffusion and application to parameter estimation. Stochastic Process. Appl. 116 1660–1675.
  • [10] Durham, G.B. and Gallant, A.R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econom. Statist. 20 297–338.
  • [11] Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959–993.
  • [12] Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177–191.
  • [13] Fearnhead, P. (2008). Computational methods for complex stochastic systems: A review of some alternatives to MCMC. Stat. Comput. 18 151–171.
  • [14] Gasbarra, D., Sottinen, T. and Valkeila, E. (2007). Gaussian bridges. In Stochastic Analysis and Applications. Abel Symp. 2 361–382. Berlin: Springer.
  • [15] Hida, T. and Hitsuda, M. (1993). Gaussian Processes. Translations of Mathematical Monographs 120. Providence, RI: Amer. Math. Soc.
  • [16] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
  • [17] Lin, M., Chen, R. and Mykland, P. (2010). On generating Monte Carlo samples of continuous diffusion bridges. J. Amer. Statist. Assoc. 105 820–838.
  • [18] Liptser, R.S. and Shiryaev, A.N. (2001). Statistics of Random Processes. I, expanded ed. Applications of Mathematics (New York) 5. Berlin: Springer.
  • [19] Mitrinović, D.S., Pečarić, J.E. and Fink, A.M. (1991). Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications (East European Series) 53. Dordrecht: Kluwer Academic.
  • [20] Papaspiliopoulos, O. and Roberts, G. (2012). Importance sampling techniques for estimation of diffusion models. In Statistical Methods for Stochastic Differential Equations. Monogr. Statist. Appl. Probab. 124 311–340. Boca Raton, FL: CRC Press.
  • [21] Roberts, G.O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm. Biometrika 88 603–621.
  • [22] Stuart, A.M., Voss, J. and Wiberg, P. (2004). Fast communication conditional path sampling of SDEs and the Langevin MCMC method. Commun. Math. Sci. 2 685–697.
  • [23] van der Meulen, F.H. and Schauer, M. (2015). Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Available at arXiv:1406.4704.